We define and study the noncommutative spectral flow for paths of regular selfadjoint Fredholm operators on a Hilbert C∗-module. We give an axiomatic description and discuss some applications. One of them is the definition a noncommutative Maslov index for paths of Lagrangians which appears in a splitting formula for the spectral flow. Analogously we study the spectral flow for odd operators on...

A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincaré duality is also shown in this...